A Modular Gradient-Sensing Network for Chemotaxis in Escherichia coli Revealed by Responses to Time-Varying Stimuli

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1 Modular Gradient-Sensing Network for Chemotaxis in Escherichia coli Revealed by Responses to Time-Varying Stimuli The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Shimizu, Thomas S., Yuhai Tu, and Howard C. erg. 2. modular gradient-sensing network for chemotaxis in revealed by responses to time-varying stimuli. Molecular Systems iology 6(): 382. Published Version doi:.38/msb.2.37 Citable link Terms of Use This article was downloaded from Harvard University s DSH repository, and is made available under the terms and conditions applicable to Open ccess Policy rticles, as set forth at nrs.harvard.edu/urn-3:hul.instrepos:dash.current.terms-ofuse#op

2 Molecular Systems iology 6; rticle number 382; doi:.38/msb.2.37 Citation: Molecular Systems iology 6:382 & 2 EMO and Macmillan Publishers Limited ll rights reserved / modular gradient-sensing network for chemotaxis in Escherichia coli revealed by responses to time-varying stimuli Thomas S Shimizu,3, Yuhai Tu 2 and Howard C erg, * Department of Molecular and Cellular iology, Harvard University, Cambridge, M, US and 2 T. J. Watson Research Center, IM, Yorktown Heights, NY, US 3 Present address: FOM Institute for tomic and Molecular Physics (MOLF), Science Park 4, 98 XG msterdam, The Netherlands * Corresponding author. Department of Molecular and Cellular iology, Harvard University, 6 Divinity venue, Cambridge, M 238, US. Tel.: þ ; Fax: þ ; hberg@mcb.harvard.edu Received ; accepted 7.5. The Escherichia coli chemotaxis-signaling pathway computes time derivatives of chemoeffector concentrations. This network features modules for signal reception/amplification and robust adaptation, with sensing of chemoeffector gradients determined by the way in which these modules are coupled in vivo. We characterized these modules and their coupling by using fluorescence resonance energy transfer to measure intracellular responses to time-varying stimuli. Receptor sensitivity was characterized by step stimuli, the gradient sensitivity by exponential ramp stimuli, and the frequency response by exponential sine-wave stimuli. nalysis of these data revealed the structure of the feedback transfer function linking the amplification and adaptation modules. Feedback near steady state was found to be weak, consistent with strong fluctuations and slow recovery from small perturbations. Gradient sensitivity and frequency response both depended strongly on temperature. We found that time derivatives can be computed by the chemotaxis system for input frequencies below.6 Hz at 22C and below.8 Hz at 32C. Our results show how dynamic input output measurements, time honored in physiology, can serve as powerful tools in deciphering cell-signaling mechanisms. Molecular Systems iology 6: 382; published online 22 June 2; doi:.38/msb.2.37 Subject Categories: signal transduction Keywords: adaptation; feedback; fluorescence resonance energy transfer (FRET); frequency response; Monod Wyman Changeux (MWC) model This is an open-access article distributed under the terms of the Creative Commons ttribution Noncommercial Share like 3. Unported License, which allows readers to alter, transform, or build upon the article and then distribute the resulting work under the same or similar license to this one. The work must be attributed back to the original author and commercial use is not permitted without specific permission. Introduction Signaling networks are complex, but typically possess modular architectures (Hartwell et al, 999). This observation provides hope for understanding the function of the whole in terms of simpler representations of its constituent parts. Much recent work on signaling networks has focused on assigning discrete functions to these parts; however, an essential next step is to determine the nature of the coupling between them. Here, we have combined theoretical modeling with in vivo measurements of signaling to characterize the transfer functions of the two modules involved in computations of time derivatives by the Escherichia coli chemotaxis-signaling pathway. The first module, which we call the receptor module, detects changes in environmental conditions to generate an intracellular signal. This module has been shown to amplify signals over a wide dynamic range (ray, 998; Sourjik and erg, 22b). The second module, which we call the adaptation module, is known to maintain the intracellular signal at a steady state that is indifferent to ambient concentrations of ligand (erg and Tedesco, 975; Spudich and Koshland, 975; arkai and Leibler, 997; lon et al, 999). oth modules have been characterized extensively by experiment, and theoretical efforts to explain their behavior have enjoyed remarkable success. However, understanding how these modules combine in living cells to produce the biologically important function of gradient sensing requires additional knowledge about the way in which they are coupled in vivo. To define this link quantitatively, we used a simple theoretical model of the chemotaxis pathway that abstracts many of the known molecular details, but preserves the essential characteristics of the receptor response and adaptation dynamics (Tu et al, 28). We exploited an important result of this analysis that provides us with a method for inferring the dynamics of one module through input output measurements of the other. Specifically, we used fluorescence & 2 EMO and Macmillan Publishers Limited Molecular Systems iology 2

3 resonance energy transfer (FRET) to monitor the real-time activity of the sensory receptor kinase complex, while subjecting cells to time-varying stimuli that render the output of this receptor module constant in time. Such a stimulus was identified by lock et al (983), who found that the output, as inferred by the switching statistics of the flagellar motor, reached a steady value during exponential ramps, that is, ligand profiles that increase or decrease exponentially in time. Under these conditions, the feedback signal must exactly cancel the change in input signal, so we can infer the dynamics in the adaptation module through measurements of the receptor module, given a well-calibrated model of the latter and good control over the temporal profile of ligand input. Fortunately, the receptor module has been characterized in detail by a large body of in vitro experiments (Dunten and Koshland, 99; orkovich et al, 992; Li and Weis, 2; ornhorst and Falke, 2; ntommattei et al, 24; Lai et al, 25), in vivo FRET experiments (Sourjik and erg, 22b, 24; Sourjik et al, 27; Endres et al, 28), and theoretical modeling (Shi and Duke, 998; Duke and ray, 999; Mello and Tu, 23, 25, 27; Shimizu et al, 23; Mello et al, 24; Sourjik and erg, 24; Keymer et al, 26; Skoge et al, 26). Our model uses a variant of the Monod Wyman Changeux (MWC) model for the receptor module, which we calibrated by measuring FRET responses to step stimuli in a series of adaptation-deficient mutants with systematically varying receptor-modification levels. This open-loop transfer function of the receptor module defines quantitatively the way in which ligand concentration, the input signal, is balanced by receptor covalent modification, the feedback signal. This model-driven approach provided two important advantages. First, we were able to measure the downstream adaptation dynamics through the same FRET method earlier developed to characterize the upstream receptor kinase response (Sourjik et al, 27). Second, because we measured the receptor module s output as a proxy for inferring the dynamics of the adaptation module, the measurement automatically yielded a second input output relation, namely the response in adaptation kinetics to input from the receptor module. In short, we could elucidate the essential dynamical character of the downstream adaptation module without directly monitoring any of its molecular constituents. In addition to ramp responses, we also measured responses to oscillatory input signals, which revealed the frequency response of the system. Together, these measurements allowed us to quantify two biologically important performance characteristics of the E. coli chemotaxis system, namely the output of time-derivative computations, and the frequency band over which such computations can be carried out. In contrast to earlier results obtained by monitoring motor switching statistics, no discontinuities in kinase activity were observed. Data collected at two different temperatures suggest a strong dependence of gradient-sensing performance on the ambient temperature. Results We present in Figure a molecular view of the chemotaxis network, and in Figure, the simplified view adopted in our ttractant ([L]) Input [L](t) m- m- (a) m+ R Receptor module m(t) G([L], m) dm dt -P F(a) Y-P a(t) daptation module Output Figure modular gradient-sensing network. () Molecular view of the chemotaxis network. The linear path from input to output begins with the input ligand concentration, [L], being sensed by the membrane-associated receptor kinase complex,, to regulate its autophosphorylation-activity, a. then transfers phosphate to the response regulator, CheY (Y), the phosphorylated form of which (Y-P) interacts with the flagellar motor (M), to control swimming behavior. The feedback loop is closed by the methyltransferase CheR (R) and the methylesterase/deamidase Che (), by regulation of the receptor methylation level, m. CheZ (Z), the phosphatase for CheY-P, decreases the signal lifetime, thus accelerating the response of the pathway. () Modular view of the network. Focusing on the functional modules, rather than the variables, of the network yields this block diagram, in which the variables are viewed as inputs or outputs (represented along wires) of two discrete signal processing modules (represented as boxes). The input output relation of the receptor module is described by the function G, which takes [L] and m as inputs to produce an output a, which connects to the downstream linear pathway toward motor output. The adaptation module constitutes the feedback loop of the network, in which the output a is converted through F(a) todm/dt and integrated over time. model. t the molecular level, the intracellular signal pathway comprises multiple species of transmembrane receptors (also known as methyl-accepting chemotaxis proteins), a scaffolding protein CheW (not shown), the histidine kinase Che, the response regulator CheY, its phosphatase CheZ, and the receptor-modification enzymes CheR and Che. The phosphorylated form of CheY, CheY-P, is the final output of the intracellular pathway that directly interacts with the flagellar motor to modulate swimming behavior. The steady-state level of this signal is determined by the balance between production of CheY-P, catalyzed by Che, and its destruction, catalyzed by CheZ. We denote the activity of the receptor kinase complex by a, which is feedback regulated by the methyltransferase CheR and the methylesterase/deamidase Che through the receptor methylation level, m. The latter is defined as the average number of methylated glutamyl residues per receptor monomer each receptor has a fixed number of these sites subject to reversible modification by CheR and Che; for the aspartate receptor Tar, there are four, so the maximum value per monomer Mmax(m)¼4. Whereas the receptor kinaseactivity a is known to increase with m, that is qa/qm4, increased a in turn leads to up-regulation of Che (orczuk et al, 986), which removes methyl groups, and downregulation of CheR (oldog et al, 26), which adds methyl groups. Thus, the sense of the feedback is negative, tending to restore a toward its steady-state value on changes in the input ligand concentration, [L]. CheZ, the phosphatase for CheY-P, Z M 2 Molecular Systems iology 2 & 2 EMO and Macmillan Publishers Limited

4 decreases the signal lifetime, thus accelerating the response of the pathway. In our FRET assay, the energy transfer pair is CheY and CheZ, fluorescently labeled by genetic fusion to yellow fluorescent protein (YFP) and cyan fluorescent protein (CFP), respectively. The FRET signal is thus proportional to [CheY-P-CheZ], the concentration of the intermediate species in the enzymatic hydrolysis of CheY-P. t steady state, the production rate of CheY-P, catalyzed by Che, is exactly balanced by its destruction rate, which is proportional to [CheY-P-CheZ]. Hence, the FRET signal serves as a readout of Che kinase activity, the output of the receptor module. In addition to the molecular constituents, we highlight in Figure three important dynamical variables from this network, namely [L], a, and m, which, respectively, correspond to the input, output, and feedback signals. We capture the dynamics of these variables in two simple equations: dm dt ¼ FðaÞ a ¼ Gð½LŠ; mþ s the receptor-modification reactions are much slower than all other reactions in the system, we model the rate of change of the feedback signal m explicity by a differential equation (equation ()), and denote its dependence on the current signal output, a, by the function F. The receptor kinase output, a, is known to relax much more rapidly (Sourjik and erg, 22a), and so its value at every instant is given by the algebraic equation (equation (2)), in which the function G denotes the dependence on the current ligand input, [L], and feedback signal, m. Thus, equations () and (2) describe the adaptation and receptor modules, respectively. This twomodule architecture is emphasized in the block diagram of Figure, which clarifies the function of the two functions appearing in equations () and (2) within this signaling circuit: F is a transfer function contributing to the feedback gain of the network, whereas G combines the ligand input [L] and the feedback signal m to produce the network output a. We use for G an allosteric model of the MWC type (Monod et al, 965), in which the effects on kinase activity of both the ligand input [L] and the feedback signal m contribute additively to the free-energy balance governing the two-state output of receptor kinase complexes with N ligand-binding subunits. The properties of the functions F and G would depend on details of the many molecular components, but, importantly, each has only one or two dynamical inputs and both have a single output. The integral sign appearing before F in Figure signifies integral feedback (Yi et al, 2), which is a wellknown engineering design for robust adaptation, and is a direct consequence of equation (): as the function F defines the feedback signal m s rate of change in time, m itself must be the time integral of F, that is mðtþ ¼ R FðaÞdt. Thus, the function F is an important design feature of this signaling circuit, as it not only determines the dynamics of the feedback regulation (equation ()), but also defines a crucial link between the two modules of the network: from the amplified output of the receptor module, to integral feedback in the adaptation module. Here, we experimentally probe this crucial link and quantitatively determine the feedback strength by measuring the response to time-varying stimuli in vivo. ðþ ð2þ Exponential ramps shift the steady-state kinase activity We first confirmed that the kinase output during exponential ramps of the form [L](t)¼[L] e rt as measured by FRET reaches a constant steady-state level. To dynamically control the input [L](t), we fabricated a mixing apparatus (described in Materials and methods) based on the design of lock et al (983). Cells were first adapted to a baseline concentration, [L], of the attractant a-methyl-dl-aspartate (Mesp). The FRET signal was stabilized at FRET, although some fluctuations were unavoidable because of noise originating in the syringe-pump-driven injection of attractant to the mixing chamber. On applying both up ramps (r4; Figure 2, C, E, and G), and down ramps (ro; Figure 2, D, F, and H), the FRETsignal was found to relax to a new, constant value, FRET c, as expected. s the magnitude of this change, DFRET (expressed in arbitrary units), is proportional to the change in kinase-activity, Da, and the latter, by definition lies on the unit interval, we can convert from FRET units to kinaseactivity (a) units by measuring the full range of DFRET. We obtain this by measuring the saturating values for DFRET in both directions, negative (DFRET sat ) and positive (DFRET þ sat ), by addition and removal of a large concentration of attractant, respectively. Thus, the pre-stimulus kinase activity a ¼ DFRET sat /(DFRET þ sat DFRET sat ), the change in kinase activity Da¼DFRET/(DFRET þ sat DFRET sat ), and the constant kinase activity during exponential ramps a c ¼a þ Da c. Ramp responses determine the feedback function and the gradient sensitivity The constant response in kinase-activity a c that is reached during exponential ramps (Figure 2 H) can be viewed as the output of time-derivative computations by the chemotaxis network. s the receptor module responds to the logarithmic change of the input signal (Tu et al, 28; Kalinin et al, 29), the relevant time derivative is that of the logarithm of input, that is dln[l]/dt, which corresponds in these experiments to the exponential ramp rate r. We therefore conducted exponential ramp-response measurements of the type depicted in Figure 2 and over a range of ramp rates r. The asymptotic kinase response, a c, obtained through such measurements, is plotted in Figure 3 as a function of r. The steady-state activity in the absence of stimuli (i.e. at r¼) was found to be a E/3. This plot reveals the sensitivity of E. coli to temporal gradients of Mesp, and the overall shape is sigmoidal, with a steep slope (Da c /DrE 3 s) near r¼. This implies that the system is tuned to respond sensitively to very shallow gradients, but it has a relatively narrow dynamic range: at greater absolute ramp rates, it becomes largely insensitive to changes in the gradient. If we define the slope Da c /Dr as the gradient sensitivity, its value is large and nearly constant in the small interval near r¼, but decays rapidly outside of it. Importantly, we observed no response thresholds at small ramp rates (Figure 3, inset), in contrast to lock et al (983), in which it was found that the ramp-response magnitude reached zero at low ramp rates (re.5 for up ramps, re. for down ramps). & 2 EMO and Macmillan Publishers Limited Molecular Systems iology 2 3

5 C D.5 ΔFRET E F G H Δa c (r) Time (s) Figure 2 Temporal profiles of pathway activity during exponential ramps. FRET responses (gray points) were observed during stimulation by exponential ramps in concentration of the form [L](t)¼[L] e rt of the attractant Mesp (blue curves). Responses were quantified by fitted functions to the FRET response (red curves). For both ramps up (r4;, C, E, G) or down (ro;, D, F, H), the FRET readout reached a steady level after an initial transient, which was well fit by a singleexponential decay. The change in the level of FRET could be converted to units of kinase activity, Da (see text), which was negative for up ramps, and positive for down ramps, as expected. The traces in the panels here were responses to ramp rates r¼±. (, ), r¼±.5 (C, D), r¼±. (E, F), and r¼±.2 (G, H). The gray points in each panel are aligned averages of two to three separately measured responses to identical stimuli. Source data is available for this figure at msb. The data of Figure 3 can be used further to infer the shape of the feedback transfer function F(a). The recorded values of a c are the outputs of the receptor module at which the effects of the time-varying ligand input, [L](t), and that of the adaptation feedback, m(t), are canceling one another exactly. Therefore, given a model of how these signals are processed within the receptor module, that is if we know the form of the function G([L], m), we can infer the rate of change of the feedback signal, that is dm(t)/dt¼f(a), from the temporal input profile [L](t) that is being cancelled. For an MWC-type receptor module experiencing exponential ramp stimuli, one can show (see Materials and methods) that cancellation of the ligand- and methylation-dependent free energy yields df L /dt þ df m /dt¼r af(a c )¼, where a df m /dm is the freeenergy change per added methyl group (in units of kt). Thus, we can rescale the abscissa of Figure 3 to render it a reading of F(a c )¼r/a, and invert the axes about the point (r¼, a c ¼a ) to obtain a plot of F(a) (Figure 3). The slope near the fixed point, F (a )E., is negative and shallow, implying weak negative feedback. Frequency response measurements reveal the time-derivative computation and its bandwidth lthough the asymptotic gradient sensitivity (Figure 3) can be viewed as the output of a time-derivative computation, 4 Molecular Systems iology 2 & 2 EMO and Macmillan Publishers Limited

6 a c r a c. r. a /3 F a /3.5.5 a Figure 3 Gradient sensitivity and the feedback transfer function F(a). () y measuring exponential ramp responses in the manner of Figure 2 H over a range of ramp rates r, we constructed a gradient-sensitivity curve, relating the kinase-activity a, to the steepness of the temporal gradient experienced by cells. The results for two FRET strains considered wild type for chemotaxis (VS4, an RP437 derivative, cyan circles; TSS78, an W45 derivative, dark blue squares) were essentially identical, and collapsed on to a sigmoidal curve with a steep region near r¼ (slope of fitted line, Da/DrE 3 s). The steady-state activity in the absence of stimuli (i.e. at r¼) was found to be a E/3. The inset in () is an expanded view about the point (r¼, a¼a ), showing the absence of thresholds, at least down to r¼±. s.() Using our model, the data of () can be used to map the feedback transfer function F(a). The steady-state relation r¼af(a c ) implies that we can rescale the r axis by the constant factor a, obtained from our calibration of the receptor-module transfer function G, and invert the axes about (r¼, a¼a ) to obtain F(a). The shallow slope near this origin, F (a )E., implies weak negative feedback. The blue curve is a fit of a Michaelis Menten reaction scheme a a FðaÞ ¼V R aþk R V ðaþ aþk ; see text for interpretation of parameters. Source data is available for this figure at it does not address what happens if such gradients change over time. For a swimming bacterium in the real world, gradients will rarely be constant, so how rapidly the cell can respond to changing gradients will also be important. We therefore measured the frequency response of the network by measuring responses to oscillatory stimuli ½LŠðtÞ ¼½LŠ e L sinð2pntþ over a range of input frequencies, n (Figure 4). For small modulation amplitudes L, the response in FRET, converted again to receptor kinase-activity units, was always found to be well fit by a sinusoid of the form a(t)¼a þ cos (2pnt j D ), where the frequency n always matched that of the input modulation, whereas the response amplitude and phase delay j D were found to depend on n (see below). This is the expected linear response to sine-wave modulation, consistent with the notion that the receptor module takes the logarithm of ligand concentration (Tu et al, 28). The measured frequency dependence of the response amplitude,, and phase delay, j D, are plotted as points in Figure 4 and C. Given the measured value for F (a ) obtained in our ramp experiments, our model can be used to determine the dependence of and j D on frequency. These predictions, shown as solid curves in Figures 4 and C, are in excellent agreement with the data. This solution yields a characteristic frequency n m defining the upper limit of the frequency band over which the system is able to take time derivatives. Using the measured parameters a E/3 and F (a )E. from our ramp-response measurements above, and N¼6, a¼2 from our calibration of the receptor module, G (see Materials and methods), we obtain n m E.6 Hz. y appropriate factoring of our model solution (see Materials and methods), we also obtain a function H, the chemotaxis system s filtering properties for the time derivative of the input signal (Figure 5, green dashed curve). The shape of H clarifies the low-pass property of this network for the derivative signal: the derivative signal is passed most efficiently below n m,in which the phase delay j D approaches p/2. Temperature affects both gradient sensitivity and frequency response ll of the above measurements were made at room temperature (22C) for consistency with earlier in vivo FRET studies (Sourjik and erg, 22b, 24), but the earlier results of lock et al (983) were obtained at the higher temperature of 32C. For comparison, therefore, we also collected ramp- and frequency response data at 32C. Interestingly, the gradient sensitivity (Figure 5) was attenuated at this higher temperature, and the slope near a (E/2 at 32C) was reduced to Da/DrE s. To compare this with the gradient sensitivity found by lock et al (983), we must convert our response, Da, measured in kinase-activity units, into D/CCWS, the response in counter-clockwise-rotational bias of the flagellar motor. Using a Hill function hccwi ¼=ð þða=kþ nh Þ for the motor response with n H ¼ (Cluzel et al, 2), we find DhCCWi=Dr qhccwi qa ðda c =DrÞE55 s, which is within threefold of the lock et al result (D/CCWS/DrE2 s). However, no response thresholds were observed, even at this higher temperature (Figure 5, inset). y the same procedure used to obtain Figure 3 from 3, we can convert the gradientsensitivity data of Figure 5 to obtain the shape of the feedback transfer function F(a) at 32C (Figure 5). The frequency response of the system was also tested at 32C using exponential sine-wave stimuli, and was again found to match closely the model prediction with n m E.8, based on the slope F (a )E.3 measured at this temperature. The shape of the gradient response (Figure 5) obtained from our FRET experiments contrasts strongly from that obtained by lock et al (983) through motor-rotation measurements, in which a flat profile near r¼ had suggested the existence of response thresholds at small values of r. These differences remain unexplained, but we speculate that they are attributable to either or both of the following: (a) the difference in the output that has been measured; whereas we have & 2 EMO and Macmillan Publishers Limited Molecular Systems iology 2 5

7 .8.4 ϕ D /2πƒ ΔFRET Time (s) ν m.6 Hz max. H H max C. ϕ D π ν (Hz) Figure 4 Frequency response and the bandwidth for derivative computations. FRET responses (gray dots) during stimulation by exponential sinusoids of the form ½LŠðtÞ ¼½LŠ e L sinð2pntþ () with a frequency n equal to that of the applied stimulus (green line), and the fit by a sinusoid (red line) of the form a(t)¼a þ cos(2pnt j D ), where is the amplitude of the response and j D is the phase delay. When the amplitude () and phase (C) of responses are plotted against the driving frequency n, the data are in excellent agreement with the analytical solutions of our model (equations (3) and (4)), plotted here without any freefitting parameters the solution uses parameters obtained separately from the ramp-response experiments of Figure 2 and 3 and MWC-model parameters obtained separately in dose response experiments using step stimuli. These analytical solutions define the characteristic frequency of the response n m E.6 Hz, below which the network is able to compute time derivatives. The dashed green curve in () is the derivative-filtering function H obtained by factoring the solution of our linearized model (equation (5)). The characteristic frequency, n m determines the upper boundary of the frequency band over which time derivatives can be computed (shaded region). Presumably, the lower boundary would be determined by a noise floor, which in this figure is taken arbitrarily to be where the response amplitude (black curve in ) falls below 5% of maximum. Source data is available for this figure at measured the kinase output by FRET between CheYand CheZ, lock et al (983) measured the motor response, or (b) the amount of data collected in the two studies; whereas our FRET technique has allowed us to efficiently collect a relatively large number of data points, each averaged over hundreds of cells (59 data points, each representing an average over hundreds of cells), the gradient response of lock et al (983) was obtained using only a few cells (26 data points from 3 cells). The 6 Molecular Systems iology 2 & 2 EMO and Macmillan Publishers Limited

8 C a c.3.2. F a.5. max 22 C 32 C 22 C Fit 32 C Fit r 22 C 32 C Fit Fit ν m.6 Hz.5 a c.2.2 r ν m.8 Hz a at 32 C /2 a at 22 C /3 rise to a discontinuous motor response. Thus, (b) seems to be the most plausible explanation. Discussion We have measured the dynamical response of E. coli s chemotactic gradient-sensing circuit to time-varying chemical stimuli. Our results reveal this signaling network s sensitivity to temporal gradients, as well as its frequency-filtering properties. Using an important relationship identified by theoretical analysis (Tu et al, 28), we derived from our exponential ramp-response measurements the feedback transfer function F(a), which couples the two modules of this signaling pathway and reveals the dynamics of the adaptation module. Responses to oscillatory stimuli, the power of which has been exemplified in a number of recent studies (ennett et al, 28; Hersen et al, 28; Mettetal et al, 28), were used to characterize the frequency response of the pathway, which determines the frequency band over which the chemotaxis system can compute time derivatives (ndrews et al, 26). We found the characteristic frequency of the adaptation system to depend on temperature, but it is in general compatible with weak feedback and large noise amplitudes at steady state. The use of time-varying stimuli allowed us to probe these dynamical characteristics of the adaptation module without direct measurement of the underlying biochemical reactions of receptor methylation/demethylation. This general approach of combining theory with experiment to probe the dynamics of pathway components not directly accessible to experiment should be applicable to many systems, and aid in the effort to uncover the design principles of biological circuits (lon, 27). We discuss here the implications of our findings in the context of earlier studies, and highlight possible directions for future work ν (Hz) Figure 5 Effects of temperature on sensitivity to gradients and frequency response. () Sensitivity to gradients was markedly decreased at 32C (red triangles; strain TSS78), in which the steady-state activity a E/2. For comparison, the data at 22C (same as Figure 3) also are plotted (cyan circles; strains VS4 and TSS78). The slope of the linear fit to the 32C points near a (red curve) was Da/DrE s. () The map of F(a) obtained by conversion of the data in () has a similar shape as that at 22C, but the slope at the zero crossing, F (a )E.3, is approximately threefold steeper, implying stronger negative feedback. The red curve is a fit to the same Michaelis Menten model as in Figure 3 (see text for parameter values and interpretation). (C) The frequency response is also shifted at 32C (red triangles). The characteristic cutoff frequency n m E.8, obtained from the model fit (black curve), is approximately threefold higher than that at 22C. For comparison, the 22C data and the corresponding model prediction from Figure 4 also are reproduced here (cyan circles and blue curve). Source data is available for this figure at dependence of motor switching on [CheY-P] was found by Cluzel et al (2) to be monotonic, steep, and highly reproducible between separately tested individual motors; therefore, there does not seem to be any way in which the continuous variation in [CheY-P] measured by FRET can give Effects of cell-to-cell variability: measurements in single cells and populations crucial difference between the measurements reported here and those of lock et al (983) is that our FRET signals are collected simultaneously from hundreds of cells, whereas lock et al recorded the motor response of individual tethered bacteria. lthough our data have much higher signal-to-noise ratios than the stochastic binary time series collected in the motor-response experiments, it is pertinent to ask whether the population-averaged nature of our FRET measurements might account for some of the differences in our observations. In particular, a striking difference was found in the exponential ramp responses to very shallow gradients (Figure 3, inset); whereas lock et al could not detect responses to ramp rates in the interval r(., þ.5) s, no such thresholds were found in the exponential ramp responses we measured by FRET. Could this difference arise from ensemble averaging the output of individual cells, which individually possess such response thresholds? We argue here that this is not possible if the typical single cell does indeed possess such response thresholds at the level of the receptor kinase response. In the context of our preceding analyses, the qualitative question of threshold & 2 EMO and Macmillan Publishers Limited Molecular Systems iology 2 7

9 existence can be recast as a quantitative one of gradient sensitivity: does there exist a region of negligible gradient sensitivity at very low ramp rates (Da c /DrE near r¼)? To make the distinction between individuals and populations explicit, we denote by a i c (r) the steady-state activity of each individual cell in a population during an exponential ramp with rate r, and by /a c S(r) the population-averaged activity we measure by FRET. s the latter is just a linear combination of single-cell activities, Dha c iðrþ ¼ P N cells Da i c ðrþ, where N cells is the number of cells in the population, the i gradient sensitivity at the population level is just the average of the single-cell gradient sensitivities Dha c i=dr ¼ N cells Pi Dai c =Dr. Thus, if the majority of cells possess response thresholds, this should have been evident in our population measurements. Transient time of ramp responses: inference of cooperativity from dynamics The gradient sensitivity, which we defined using the constant kinase-activity a c reached during exponential ramps, can be viewed as the output of time-derivative computations by the chemotaxis system. Remarkably, the solution for this output in our model, a c ¼F (r/a), is independent of the parameter N, which represents the degree of receptor cooperativity in the MWC model. Thus, although receptor interactions have an amplifying function in the open-loop response to step and impulsive stimuli (Sourjik and erg, 24; Keymer et al, 26; Mello and Tu, 27; Tu et al, 28) the property traditionally referred to as the gain of the chemotaxis system (cf. e.g. Segall et al, 986) the amplitude of responses to sustained temporal gradients is dictated only by the adaptation system (through the function F(a)). This does not mean, however, that receptor cooperativity is unimportant within the closed-loop control structure of the chemotactic gradient-sensing circuit. The function of receptor coupling becomes clear when one considers the time required for the chemotaxis network to take this time derivative, which is an equally important measure of performance for this gradient-sensing system. Our model provides a basis for investigating the molecular parameters that determine the speed of this computation, and yields the following simple, approximate expression for the time, t, required for the kinase activity to reach a c during exponential ramps: t Dfða c Þ=Nr where Df(a c ) is the free energy difference between the steady states with and without the ramp: Df(a c )¼ln(/a c ) ln(/a ). The quantity Df(a c )/N can be intuitively understood as a characteristic amount of chemical work that the adaptation system must perform before it can catch up with the rate r at which the ligand concentration is being ramped. In the limit of shallow ramps (small r), where Da c is small, we can linearize equations () and (2) to solve for a limiting, minimum response time, t -t m ¼( ana ( a ) F (a )) ¼ (2pn m ), where n m is the characteristic frequency of the linear response to oscillatory stimuli (see equation (4) of Materials and methods). This solution is independent of the applied (small) ramp rate r. t greater ramp rates, the linear approximation is no longer valid, but we can still estimate t by the simple relation presented in equation (3). The justification for this approximation is illustrated in Figure 6, which is a plot of the time course of the two free-energy contributions to kinase activity, stemming from ligand binding (f L ) and methylation (f m ). When an exponential ramp with rate r is started at t¼, f L begins to rise immediately as rt. The methylation-dependent contribution, f m, lags behind f L, but this lag asymptotically stabilizes as the methylation rate F(a) reaches r/a, from which point on, the free-energy lag ð3þ Kinase activity, a a(t)-a c ~ e t/t a a c.5 rt Free energy, ƒ Δƒ(a c )/N t ƒ L (t) ~ rt ƒ m (t) ~ αf(a)t rt Δƒ(a c ) Time, t Figure 6 Transient responses to ramps: the time required for derivative computations. () Schematic illustration of the changes in activity and free energies during exponential ramps. s the receptor kinase activity approaches the constant value a c during exponential ramps, the rate of change of ligand- and methylation-dependent free energies (f L and f m, respectively) balance one another, leading to the asymptotic relation r¼af(a c ) between the exponential ramp rate r and the net rate of change in intracellular methylation F(a) (see also Materials and methods). The time required for f m (t) to change by an amount equal to the total free-energy change Df(a c ) can be estimated as t EDf(a c )/Nr, and provides a robust estimate for the time required for the activity a(t) to reach a c (see text). () plot of rt versus Df(a c ) for exponential ramp responses at 22C. The slope of the fitted curve yields an estimate for the extent of receptor cooperativity, NE4.9, which is in good agreement with the value of N¼6 obtained independently from dose response curves from experiments using step stimuli (see text). 8 Molecular Systems iology 2 & 2 EMO and Macmillan Publishers Limited

10 Df(a), and hence the kinase activity become constant (i.e. Df (a)-df (a c ), as a(t)-a c ). In the linear regime in which the transient time is independent of r, a(t) will decay exponentially in time as e t=t toward a c, and equation (3) is exact. The constant solution t ¼t m in this case can be obtained by letting r¼af (a )(a c a ) in equation (3) and taking the limit a c -a. In the more general case, t will not be independent of r, but equation (3) still provides a useful approximation. In particular, it retains the correct scaling with the extent of receptor cooperativity, N. To compute the points in Figure 6, t was estimated by fitting an exponential decay function of the form aðtþ ¼a c þða a c Þe t=t to the FRET data recorded during exponential ramp stimuli (such as those of Figure 2) at different values of r, and Df(a c ) was computed from the a and a c values from the same measurement. s Df (a c ) is independent of the degree of receptor cooperativity, N, we can use equation (3) to extract from our experimental data an estimate for the parameter N. lthough it recently has been reported (Endres et al, 28) that the degree of receptor cooperativity seems, under certain conditions, to depend on the modification level of receptors (i.e. that N could depend on m, which is changing during our ramp measurements), the fact that we observe a constant activity during exponential ramps suggests that such changes in the organization of the receptor complex occur on time scales considerably slower than that of our time-varying stimuli. We, therefore, apply equation (3) to infer a single value of N from the set of t values extracted from our ramp-response data. lthough accurate determination of t is difficult, because its estimation is more sensitive to experimental noise, the approximate linear scaling of the product rt N Df (a c ) follows immediately from equation (3), and is more robust to noise than t itself Figure 7 Calibration of the receptor-module transfer function, G([L],m). () FRET responses in kinase activity (points) to step stimuli of the attractant Mesp were measured in a series of mutants in which the modification state of Tar receptors were fixed by deletions of the cher and che genes. To reveal the response of the Tar receptor, these measurements were conducted in a genetic background in which Tsr and Tap receptors are not expressed. Fits to the allosteric MWC model (equations (6) (8)), with parameters N¼6, K I /K ¼.62 are shown as solid lines. The gray level of the points and solid curves indicate the modification state of the mutant strain, and is tabulated in the legend (lighter shades of gray for higher modification levels). The blue points and curve, denoted wt in this figure, is the strain VS78, which has the same receptor complement as the other mutants (Tsr Tap ), but retains the wild-type genes for CheR and Che. Kinase activity is shown normalized to the pre-stimulus value for the strain VS4. () Values of f m, obtained for each modification state represented in (), plotted against the number of modified sites. The data for modification states containing only glutamate (E) and glutamine (Q) residues (black points) fell on a straight line (dotted) when plotted as a function of the number of E-Q transitions (n E-Q ; black points). The data for modification states containing only Q s and methylated glutamates (Em) fell on a different straight line (dashed), when plotted as a function of the number of Q-Em transitions (n Q-Em ; magenta points). The values for the two extreme methylation levels, EEEE (corresponding to m¼) and EmEmEmEm (corresponding to m¼4), were obtained by extrapolation of the two straight lines, as they could not be obtained directly from fits to the FRET data (cells with the Tar population fixed in this state did not respond to any concentration of Mesp, as is seen in ()). The solid line connecting these two extreme states reveals the dependence of f m on the E- Em transitions (n E-Em ; gray points), and the CheR þ Che þ strain VS78 (blue point, denoted wt) falls on this line, as expected. When fitted by equation (9), this line yields the parameters m E.5 and ae2 kt used throughout this study. Source data is available for this figure at (as the factor r conveniently diminishes the error more strongly in which Da c is small, and errors in estimating t tend to be large). In Figure 6, we plot rt against Df (a c ), in which the slope of the fitted line gives an estimate N4.9, which is in satisfactory agreement with the more reliable value of N¼6, estimated through fits to dose response curves to step stimuli (Mello and Tu, 27; see also Materials and methods and Figure 7). iochemical interpretation of F(a): enzyme saturation and Che activation The steady-state identity r¼af(a c ) derived from our model allowed us to map the feedback transfer function F(a) from the gradient-response data measured at each temperature. We found that F(a) is a nonlinear, monotonically decreasing function with a shallow slope near the steady-state fixed point, F(a )¼, and a much steeper slope near the high-activity extreme, F(). s F(a) represents the net rate of change of Kinase activity (relative) ƒ m (m) / kt.5.5 EEEE wt QEEE QEQE QEQQ [Mesp] (mm) QQQQ QEmQQ QEmQEm EmEmEmEm EEEE wt QQQQ QEEE QEmQQ QEmQEm QEQE QEQQ QQQQ EmEmEmEm n E Q, n Q Em, or n E Em & 2 EMO and Macmillan Publishers Limited Molecular Systems iology 2 9

11 methylation catalyzed by CheR and Che, a biochemical interpretation of this curve can be obtained by an enzymatic reaction model, a FðaÞ ¼V R K R þ a V a ðaþ ð4þ K þ a where K R and K are the Michaelis constants for each reaction, and V R and V (a) represent the velocities of the methylation and demethylation reactions, respectively, when those enzymes are saturated with substrate (all concentrations are normalized, in units of the Che kinase concentration; for example K R, are dimensionless, V R, have units s ). This expression assumes that CheR can only bind the inactive fraction and Che can only bind the active fraction of receptor kinase complexes. Our measurements place strict limits on these end point values, F() and F(). In our rampresponse data, F() can be inferred from the ramp rate r c at which the response amplitude to positive ramp-stimuli saturates (i.e. in which a c reaches zero) by the relation r c ¼aF(). Reading off values for r c from Figure 5, and using again a¼2 from our receptor-module calibration (described in Materials and methods), we expect F()E. at 22C and F()E.25 at 32C. The data are noisier at the opposite extreme of activity, but an approximate lower bound for the absolute value of F() can readily be inferred by examining the maximal absolute response observed, which yields F() 44F() at both temperatures. These constraints, combined with the high curvature near F() and the location of a, precluded a fitting with a constant value for V. Evidently, what is required is a sharp increase in V (a) as a approaches unity. The solid curves shown in Figures 3 and 5 use a simple piece-wise linear form, V ðaþ ¼V ðþð þ yða a Þ a a a r Þ, where y(x) is the unit step function (y(x)¼ for x4, y(x)¼ otherwise), a represents the value of a above which a V (a) increases, and r determines the maximum value of V (a). For the data of Figure 5, we found that a ¼.74 yields good fits for data at both temperatures. Values for other parameters are as follows: at 22C, {V R,V ()}¼{.,.3} s, {K R, K }¼{.32,.3}, and r ¼4.; at 32C, {V R,V ()}¼{.3,.3} s and {K R, K }¼{.43,.3}, and r ¼2.7. The comparison between the two sets of experiments offers some insight regarding the temperature dependence of the parameters. The dominant difference is in the saturating velocities for methylation and demethylation, V R and V (a), which can be interpreted as the product of the active enzyme concentration and the catalytic rate constant k cat. s the growth conditions before the experiments were identical, and measurements were conducted in a medium that does not support protein synthesis, the expression levels of enzymes were the same in the experiments at 22 and 32C. Thus, we conclude that k cat s for methylation and demethylation are the parameters most sensitive to temperature in the adaptation system. The mechanism responsible for the strongly nonlinear behavior of V (a) remains to be determined. likely candidate is the phosphorylation of Che, but we note that straightforward mechanistic assumptions, such as a linear, hyperbolic, or quadratic dependence of [Che-P] on a would not suffice to produce the observed sharp increase of V (a)asaapproaches unity. Future experiments with phosphorylation-deficient mutants could shed light on this issue. Noise and feedback dynamics near steady state Single-cell measurements of motor output (Korobkova et al, 24) have established that the steady-state activity of the E. coli chemotaxis pathway exhibits large fluctuations over a slow timescale. Through theoretical modeling and simulations, Emonet and Cluzel (28) have argued that this could be attributable to the adaptation enzymes CheR and Che operating near saturation, a phenomenon related to zeroorder ultrasensitivity of reversible covalent-modification systems (Goldbeter and Koshland, 98). Our data provide an experimental test for this hypothesis, as the biochemical parameters inferred from the above analysis of F(a) can be used to assess the degree of saturation of the in vivo adaptation kinetics, and, in turn, the steady-state noise. The values we obtained above of the Michaelis constants, K R and K, and the pre-stimulus kinase-activity a, imply that 47% of both CheR and Che are bound to their receptor substrates at steady state. Here, we apply the stochastic analysis of Emonet and Cluzel (28) to our biochemical model for the adaptation system (equation (4) of the main text), to estimate the steady-state noise expected for kinase activity in vivo. The feedback transfer function F(a) determines the dynamics of the adaptation system through equation (). In the absence of chemical stimuli, the pathway activity, in turn, is determined completely by the adaptation system; by taking the time derivative of equation (2) and substituting into this equation (), we find da dt ¼ da dm dm dt ¼ ana ð a ÞFðaÞ. pplying this result to the stochastic time-evolution equation obtained by Emonet and Cluzel (28) by the linear noise approximation, the displacement from the steady-state activity, Da(t)¼a(t) a, evolves in time as ddaðtþ dt ¼ t a pffiffiffi DaðtÞþ D ZðtÞ where t a ¼( ana ( a )F (a )) is the relaxation time of the receptor kinase-activity a, Z(t) represents temporally uncorrelated white noise, and D ¼ da 2=Ntot a dm ðv a R a þk R þ V ða Þ a þk Þ is the strength of noise originating from the discrete methylation events occurring in N tot receptor complexes. Computing the value of the relaxation time constant t a using the parameters a, F (a ), N and a from our measurements, we obtain t a E29 s at 22C and t a E s at 32C. The solution for the amplitude of the steady-state fluctuations, s a, in the Emonet Cluzel model p (equation (5)) is s a ¼ ffiffiffiffiffiffiffiffiffiffiffiffi t a D=2. With parameters for the CheR/ Che reactions inferred from our measured profile of F(a), and assuming N tot ¼ 4 /N (i.e. partitioning the 4 receptor dimers in each cell into MWC clusters of size N), we obtain s a /a E.87 at 22C and s a /a E.77 at 32C. Thus, our results are consistent with slow steady-state fluctuations in kinase activity, with a large amplitude of around % of the mean and relaxation time constants in the range 3 s. In an analysis of the noise-spectral data obtained by Korobkova et al (24), Tu and Grinstein (25) ð5þ Molecular Systems iology 2 & 2 EMO and Macmillan Publishers Limited